How to get a material to keep the transparent border from it's texture
Here I have added a Color/Invert Node to swap the Black/White of your texture before feeding it into the Color of the Mix Shader





On the left is with the Invert node and on the right without. Showing what is and what is not being transparent.



and the final result.

FYI For more control plug the Alpha into the Factor on the Invert Node and then change the colour.